•NRLF 


AND  VALLEY 


DETAILS,  FORMULAE  AND  GRAPHICS 


-  .-SYLES  iOF    HIP    RAFTER    CONNECTIONS 


HIT  HIT 

IIII      Illl 


HIP  AND  VALLEY  DESIGN 


DETAILS,  FORMULAE  AND  GRAPHICS 

ROOFS 

HOPPERS  AND  PIPE  LINES 


BY 


H.  L.  McKIBBEN  and  L.  E.  GRAY 


Edited  by 

J.  E.  BANKS,  Engineer  Bureau  of  Standards 

American   Bridge  Company 


Published  by 

J.  E.  BANKS 

Ambridge.  Pa.  Price.  $2.00 

Postpaid 


Copyright,  1912,  by 

H.  L.  McKIBBBN  and  L.  E.  GRAY 

Ambridge,  Pennsylvania 


First  Edition. 

Second  Thousand,  November  1,  1913 


PREFACE 


The  difficulty  of  making  working  shop  drawings  for  roof 
connections  at  Hip  and  Valley  is  appreciated  by  Structural  Engineers. 

This  book  has  been  prepared  to  cover  practical  working  details 
for  such  construction  and  to  present  the  analytic  and  graphic  processes 
needful  for  their  development. 

From  the  presentation  of  the  designs  here  given,  Engineers  and 
Architects  can  determine  the  style  of  connection  adapted  to  their 
demands  readily  and  can  specify  the  same  for  the  structures  they  have 
in  charge. 

To  Draftsmen  the  treatment  of  the  subject  will  especially  appeal, 
resulting  to  them  in  a  saving  of  extra  labor  and  concern. 

Students  will  discover  the  practical  training  in  descriptive 
geometry  and  trigonometry  as  applied  to  active  engineering  to  be 
exceptionally  valuable.  Class  room  work  in  the  proof  of  the  formulae 
is  recommended  to  Engineering  Schools. 

H.    L.    McKlBBEN. 

L.  E.  GRAY. 

Engineers  with  American  Bridge  Co. 


282177 


FOREWORD. 


On  pages  3,  4  and  5  are  shown  working  details  for  styles  A,  B,  C,  D,  E 
and  F,  or  six  methods  of  connection  to  Hip  Rafters  from  which  to  select  the 
one  that  conforms  best  to  the  adjoining  framing. 

On  pages  6,  7  and  8  are  found  working  details  for  styles  A,  B,  C,  D,  E  and 
F,  or  six  methods  of  similar  connections  to  Valley  Rafters  from  which  to  choose 
the  one  most  desirable.  * 

A  sketch  appears  with  each  style  of  detail  showing  the  position  of  the 
purlin  in  the  main  roof  section,  and  small  sub-formulae  showing  solutions  for 
the  variables  yi  to  yio,  with  special  attention  to  yi  and  yz. 

After  making  selection  of  style  desired,  the  detailer  should  solve  the  angles 
required  as  shown  in  details;  i.  e.,  L8  and  L9  are  needed  in  style  C.  No  other 
angles  need  be  found ;  only  those  involved  in  the  style  chosen. 

Solution  of  these  angles  can  be  readily  made  from  general  formulae  on 
page  10,  if  the  worker  be  familiar  with  Trigonometry  and  Logarithms;  if  not, 
results  may  be  obtained  from  the  simple  graphics  given  on  pages  11,  12  and  13, 
making  the  problem  easy  for  the  detailer  who  is  not  familiar  with  formulae. 

If  the  case  in  hand  be  one  that  is  covered  by  the  tabulated  solutions  on 
pages  14,  15  and  16,  the  worker  can  take  from  those  tables  any  or  all  variables 
which  develop  in  a  roof  of  pitch  1/5,  V4,  1/3,  30°  or  55°,  if  the  angle  B  in  plan  is 
30°,  45°  or  50°. 

These  tabulated  solutions  give  the  values  of  the  variables  for  designs  in 
most  common  use  without  the  necessity  of  solving  any  angles  whatever;  but  the 
formulae  on  page  10  and  graphics  on  pages  11,  12  and  13  furnish  data  for  solving 
angles  for  any  roof  pitch  and  all  possible  positions  of  rafter. 

In  styles  A,  B,  C  and  E  the  roof  line  being  above  the  main  truss  metal  line, 
the  worker  will  need  to  use  formulae  on  page  9  to  locate  working  point  "d." 

The  authors  desire  to  call  especial  attention  to  the  following: 

1st.  The  known  data  are  in  all  cases  the  main  roof  pitch  or  Angle  A. 
The  position  of  Hip  or  Valley  Rafter,  Angle  B,  which  is  the  angle  formed 
by  rafter  and  main  truss  as  seen  in  Plan  looking  directly  perpendicular  to 
lower  side  of  Angle  A.  No  other  data  than  A  and  B  as  above  described 
is  ever  required. 

Throughout  both  details  and  graphics  the  letter  "d"  refers  always  to 
the  same  working  point;  the  marks  di  and  da  refer  also  to  this  same  point, 
viewed  from  different  positions. 

2d.  All  formulae  on  page  10  are  logarithmic,  and  in  terms  of  tangent 
functions. 


3d.  Use  of  the  graphics  on  pages  11, 12  and  13  expedite  the  work  and 
give  accurate  results. 

4th.  A  short  method  of  graphics  for  solution  of  Angles  L5,  L6  and  L8 
also  appears  on  page  10,  which  may  be  used  after  solving  L3  and  L4,  if 
desired. 

5th.  For  those  desiring  to  follow  out  the  proofs  given  on  pages  21  to 
29,  the  four  major  intersecting  planes  involved  are  as  follows  (see  page  10) : 

ROOF    PLANE. 

Seen  in  Elevation  of  Truss  as  line  ab. 
Seen  in  Plan  as  inclined  surface  ai,  bi,  n. 

PURLIN    WEB     PLANE. 

Seen  in  Elevation  of  Truss  as  line  c,  d. 
Seen  in  Plan  as  inclined  surface  di,  ci,  ei. 

RAFTER    WEB     PLANE. 

Seen  in  Elevation  of  Rafter  as  surface  ra,  ca,  bs. 
Seen  in  Plan  as  line  n,  bi. 

RAFTER    FLANGE    PLANE. 

Seen  in  Elevation  of  Rafter  as  line  rz,  ba. 
Seen  in  Plan  as  inclined  surface  n,  bi,  ei. 

6th.     Other  formulae  which  may  be  used  if  desired  are  as  follows: 

Cos  L3=Cos  R  Cos  L1  Sec  A. 

Tan  L5=Cos  A  Tan  B  Cos  L1. 

Tan  L5=Tan  L2  Cos  L1. 

Tan  A7=Sin  A  Sin  B  Cos  L4. 

Tan  L7=Cos  L2  Tan  L10. 

HOPPERS,    BINS   AND   CHUTES    (FORMS   OF   VALLEY   CONSTRUCTION). 

Details  for  these  structures  are  left  to  the  judgment  of  the  detailer  and 
are  usually  governed  by  the  main  design.  The  solution  of  the  bend  on 
connecting  plate  at  dihedral  intersections  is  the  only  difficulty  for  most 
draftsmen.  Both  formulae  and  graphics  are  provided  on  page  17  for  ready 
use. 

PIPE   LINES. 

Large  Pipe  Lines  often  require  both  horizontal  and  vertical  change  of 
direction  at  the  same  point,  which  condition  may  give  rise  to  annoying 
details.  Two  separate  bends  are  more  expensive  and  produce  greater 
friction  on  the  flow  than  a  single  resultant  bend.  Careful  attention  to 
resultant  angles  "X"  and  detail  angles  "Y"  will  save  much  trouble  in 
fabrication  and  improve  the  efficiency  of  the  finished  structure. 


HIP     RAFTER     DETAILS 


Xi=gwidthof  rafter  flange  or  more 
.     yi  =Xi  tanB  sinR 


JPurlln  may  be  cut  on 
atted  line  if  desired 


<  g2   x 

V 

\ 
\\          < 

•   ,    I 

i- 

.-.-..€  J 

W1 

L2 

2ut  flange  to  clear 

,IL 

/ 

STYLE  A. 


Xi  =  gwidth  of  rafter  flange  or  more 
yi  =  Xi  tan  B  sin  R 
y2=yi  tan  £3 


STYLE   B. 


HIP     RAFTER     DETAILS 


width  of  rafter  flange  or   more. 
yi  =  Xi    tan   B  sin  R 
ya=Xi  tan  B  cos  R 
y4=Xi  sec  B 


Developed  plate 


STYLE   C. 


d 

k 

1 

( 

(V4, 

Ii-y4 

? 

<84 

5 

«. 

.---- 

^ 

-''r 

i                    ( 

< 

r 

•             \ 

^ 

t           d, 

\ 

.'- 

?'       fe            J 

Xi=2  width  of  rafter  flange  or  more. 

X2,  taken  so  that  bend  line  will  clear  connection  angle. 

yi=Xi  tan  B  sin  R 

ys=X2  tan  B  cos  R 

ye=X2  sec  B 


STYLE    D.      Developed  plate 


5 


HIP  RAFTER   DETAILS 


Xi  =  2  width  of  rafter  flange  or  more, 
t  =  thickness  of  bent  plate. 

yi  =  Xi  tan  B  sin  R 

ya=Xi  tan  B  cos  R 
sec  B 


STYLE   E. 


width  of  rafter  flange  or  more, 
thickness  of  rafter  web. 

yi  =  Xi  tan  B  sin  R 

V2=*yi  tan  L3 

y4=,Xi  sec  B 

zi  =  t  sec  L5 


flange  cut  to  clear. 


STYLE   F. 


VALLEY  RAFTER  DETAILS 


-Developed  plate 


STYLE    A. 


82 


L1 


plate 


STYLE    B. 


VALLEY  RAFTER  DETAILS 


Xa  taken  so  that  Bend  Line 
will  clear  connection  angle. 
y?=.X3  tanBcosR 
secB 


STYLE    C. 


X4  taken  so  that  Bend  Line 
will  clear  connection  angle 
yg  =  X4  tan  B  cos  R 
yio  =  X4  secB 
=  Xi  secB 


STYLE    D. 


8 


VALLEY  RAFTER  DETAILS 


Xi=a  width  of  rafter  flange  or  more. 
t=thickness  of  bent  plate. 

yi  =  Xi  tan  B  sin  R 

V3=Xi  tan  B  cos  R 

X4=Xi  sec  B 


STYLE   E. 


Xi=£  width  of  rafter  flange. 
ti=£  thickness  of  rafter  web. 
t2=thickness  of  bent  plate. 

y4=Xi  sec  B 

Z2=ti   sec  L5+t2  tan  L5 


STYLE    F. 


Purlin  and  bent  plate   must 
clear  rafter  flange  as  shown. 


ivelope'd  plate 
bend  line 


LI 


RELATIONS  OF  ROOF  LINE  TO  WORKING  POINTS 
USED  IN  FORMULAE 


9 


Xi  or  Xi=5  width  of   rafter  flange  or  more. 
P=actual  depth  of  purlin. 


D=P.  sec  L10 
yi  =  Xi   tan  LIO 
D'=P'sec  LW 
yt=D+yi-D' 
Xi^y/cot  LIO' 
y2=yi  tan  L3 
y2=yi'tan  U? 
w=D  tan  L3 
w=D'tan  L3r 
N=P  tan  LI 
N  =  P'tan  Lf 


y=D-D' 
y2-y    tan  L3' 
w  =D   tan  L3 
w'=D'  tan  L3' 
N  =P   tan  L1 
N'  =  P'  tan  LT 


10 


GENERAL  FORMULAE 


Given  L3  and  L4      \]/L3 
to  find  L5.  \fl 


FIG.  3 

Given  L3  and 
to  find  L6. 


F1G.1 


FIG.  4 
Given  L3  and  L4 
to  find  LS. 


In  figs.  2, 3  and  4,  the  line  xx  is  the  intersection 
of  Rafter  Flange  and  Rafter  Web. 


GIVEN 

A=Pitch  of  Roof. 

B=Angle  between  Truss  and  Rafter 
in  plan. 

FORMULAE 
Tan  R  =tanA  cosB 
Tan  L  /=sin  A  tan  B 
Tan  Z.2=cosA  tanB 
Tan  £3=sln  A  cos  A  sin  B  tan  B 
Tan  L4=cosaA  tan  B  sec  R 
Tan  £5=cos  L3  tan  L4 
Tan  /.ff=tan  L3  cos  L4 
Tan  /L7=tanB  sin  R  cos  L2 
Tan  L8=cos  A  tan  B 
Tan  L9=tan  B  sin  R 
Tan/./^=tan  B  sin  R 


11 


GRAPHIC   SOLUTION   OF  ANGLES 


bl,CI 


A  =  PITCH  OF  ROOF 

B  =  ANCLE  BETWEEN  TRUSS  AND  RAFTER 
IN  PLAN 

R=  PITCH  OF  RAFTER 

Tan  R— Tan  A  Cos  B 


L1  —  BEVEL  ON  PURLIN  WEB  PLANE  MADE 
BY  INTERSECTION  OF  RAFTER  WEB 
PLANE 


FORMULA 
Tan  i7  = 


A  Tan  B 


GRAPHICS 

Draw  d,  c  l  a,  b 
Draw  d,  dl  |  b,  bl 
Revolve  d  to  f  ,  about  c 
Draw  f,  fl  |  d,  dl 
Draw  dl,  fl  ±  d,  dl 
Connect  fi  with  ci 


L2 — BEVEL  ON  ROOF  PLANE  MADE  BY 
INTERSECTION  OF  RAFTER  WEB 
PLANE 


FORMULA 


Tan  B  Cos  A 


GRAPHICS 

Revolve  b  to  g  about  a 
Draw  g,  g\  |  b,  bi 
Extend  ai,  bl  to  g\ 
Connect  g\  with  ri 


L3 — BEVEL  ON  RAFTER  WEB  PLANE  MADE 
BY  INTERSECTION  OF  PURLIN  WEB 
PLANE 

FORMULA 

Tan  Z.3  —  Sin  A  Cos  A  Sin  B  Tan  B 

GRAPHICS 

Draw  d,  c  i  a,  b 
Draw  d,  dl  |  b,  bl 
Draw  di,  d2  1  bi,  b2 
Connect  d2  with  C2 


D1.CI 


b,--' 


bia 


12 


GRAPHIC     SOLUTION     OF     ANGLES 


L4 BEVEL       ON       RAFTER      FLANGE 

PLANE  MADE  BY  INTER- 
SECTION OF  PURLIN  WEB 
PLANE 

FORMULA 

Tan/14  =  Cos2  A  Tan  B  Sec  R 
GRAPHICS 

Draw  d,  c  i  a,  b 

Draw  d,  di  ||  b,  bl 

Drawdi,d2  ||bi,b2 

Revolve  d2  to  h,  about  r2 

Draw  h,  hi  ||bi,  b2 

Extend  b,  bl  to  intersect  n,  r2  at  el 

Connecfel  with  hi 


L5 — COMPLEMENT  OF  ANGLE  BE- 
TWEEN PURLIN  WEB  PLANE 
AND  RAFTER  WEB  PLANE 

FORMULA 

Tan  L5  =  Cos  L3  Tan  L.4 
GRAPHICS 

Draw  d,  c  l  a,  b 

Draw  d,  dl  ||  b,  bl 

Draw  dt,d2  ||  bl,  b2 

Draw  d2,  Z2  l  b2,  02 

Draw  Z2,  zs 1  d2,  02 

Revolve  23  to  Z4  about  Z2 

Draw  Z4,  zs  II  bl,  b2 

Locate  ze  at  intersection  of  d,   dl 
and  ci,  C2 

Connect  zs  with  ze 

L6 — COMPLEMENT  OF  ANGLE  BE- 
TWEEN PURLIN  WEB  PLANE 
AND  RAFTER  FLANGE  PLANE 

FORMULA 

Tan  L6  =  Tan  L3  Cos  L.4 
GRAPHICS 

Draw  d,  c  l  a,  b 

Draw  d,  di  ||  b,  bi  and  extend  to  v 

Extend  b,  bi  to  ei 

Connect  el  with  di 

Draw  es,  V3  ||  ei,  dl 

Draw  ei,  03  and  di,  V3  l  ei,  dl 

Take  V3,  d3  =  d,  v 

Connect  es  with  d3 

Through  n,  draw  V4,  vs  l  ei,  di 

Draw  V4,  p  l  63,  d3 

Revolve  p  to  ve  about  v4 

Draw  ve,  v?  J.  ei,  di 

Connect  v?  with  n  and  vs 


13 


L7 — BEVEL  ON  PURLIN  WEB  PLANE  MADE 
BY  RAFTER  FLANGE  PLANE 

FORMULA 

Tan  Z.7      Tan  B  Sin  B  Cos  L.2 
GRAPHICS 

Draw  d,  c  i  a,  b 

Draw  d,  di  ||  b,  bi 
Revolve  d  to  f  about  c 
Draw  f,  fi  ||  b,  bi 
Draw  di,  fi  j.  d,  di 
Extend  b,  bi  to  ei 
Connect  ei  with  fi 

L8 — ANGLE  BETWEEN  PURLIN  WEB  PLANE 
AND  A  PLANE  PERPENDICULAR  TO 
BOTH  RAFTER  WEB  PLANE  AND 
RAFTER  FLANGE  PLANE 


FORMULA 
Tan  L8  = 
GRAPHICS 


Tan  B,  Cos  A 


Draw  d,  c  l  a,  b 

Draw  d,  di  ||  b,  bi 

Draw  d,  m  l  b,  c 

Draw  m,  s  l  d,  c 

Revolve  s  to  n  about  m 

Draw  n,  p  ||  d,  di 

Draw  di  p  j.  d,  di 

Draw  di,  v  l  ri,  bi  to  intersect  b,  bi  at  v 

Connect  v  with  p 

L9 — BEVEL  ON  PLANE  PERPENDICULAR 
TO  BOTH  RAFTER  WEB  PLANE  AND 
RAFTER  FLANGE  PLANE  MADE  BY 
INTERSECTION  OF  PURLIN  WEB 
PLANE 
FORMULA 

Tan  L9  =  Tan  B  Sin  R 
GRAPHICS 

Draw  d,  c  l  a,  b 

Draw  d,  di  ||  b,  bi 
Draw  di,  d2  ||  bi,  b2 
Draw  d2,  k  l  r2,  b2 
Draw  k,  ki  JL  ri,  bi 
Revolve  k  to  j  about  ,d2 
Drawj,  ji  ||  k,  ki 
Draw  ki,  ji  Ik,  ki 
Connect  d  i  withji 

L10 — ANGLE        BETWEEN       ROOF       PLANE 
AND  RAFTER  FLANGE  PLANE 

FORMULA 

Tan  L.10  =  Tan  B  Sin  R 
GRAPHICS 

Take  p  any  point  on  b2,  r2 

Draw  p,  1 1  b2.  r2 

Revolve  p  to  u  about  t 

Draw  t,  tl  ||  bl,  b2 

Draw  u,  ui  ||  bi,  b2 

Locate  a  at  intersection  of  t,  ti  and  a,  n 

Connect  ui  with  s 


bi  ci 


b c 


14 


SOLUTIONS,  FIVE  ORDINARY  ROOF  PITCHES 

B=30° 


A 

1/5  PITCH 

1/4 

PITCH 

30° 

PITCH 

1/3 

PITCH 

65°  PITCH 

>< 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

R 

L      1 
L     2 
L     3 
L     4 
L     5 
L     6 
L     7 
L     8 
L     9 
L    10 

4%2 
2%8 

6%2 

Uie 
2 

2%2 
2%2 

9.53959 
9.33127 
9.72921 
8.99801 
9.72159 
9.71945 
8.94484 
9.22157 
9.72921 
9.27642 
9.27642 

3%2 

6y82 

6 

2% 
2% 

9.63650 
9.41195 
9.71298 
9.06247 
9.70185 
9.69897 
9.01344 
9.30929 
9.71298 
9.36062 
9.36062 

6 
6 

5i%8 

5% 

6 

3%2 
3%2 

9.69897 
9.46041 
9.69897 
9.09691 
9.68496 
9.68159 
9.05119 
9.36350 
9.69897 
9.41195 
9.41195 

5% 

517/82 

3% 

58/4 
31  %2 

9.76144 
9.50550 
9.68159 
9.12462 
9.66421 
9.66039 
9.08268 
9.41532 
9.68159 
9.46041 
9.46041 

1427/32 

38y32 

1% 

3% 

31%2 

5% 
5% 

0.09230 
9.67480 
9.52003 
9.13237 
9.48016 
9.47620 
9.11340 
9.62961 
9.52003 
9.65221 
9.65221 

"S"  =  Corresponding  Bevels  or  Slopes  to  Base  of  12  inches. 

A 

1/5  PITCH 

1/4 

PITCH 

3O°  PITCH 

1/3 

PITCH 

66°  PITCH 

X   1 

iH 

2K   3   ' 

t/i 

6K 

1H 

&A 

3    ' 

fcK 

6M 

IK 

2^ 

3 

4/4 

6^ 

1K2K 

3  ' 

iKsM 

IX 

i 

Y   1 
Y  2 
Y  3 

Y  4 

T92 

A 

H 

if! 

TV     TV 
2%  3^f 

11 

A 

2i  i; 

MS 

^ 

1  1 

A 

II 
if! 

A 
iV 

2% 

H 

A 

it! 

Sg"^ 

fl 
.> 

H 

2/4 

*;! 

^•16 
5 

TV 
ft 

HI 

21 

A 
iA 

H 
A 

IT'S 

iA 

2f\ 
*f! 

1^ 

9w] 
*  57* 

TV 
TV 

if! 

If 

A 

2H 

I 

lA 
A 

iii 

7A 

1  1 

A 
H 

HI 

1/8  ^-3^ 

A     T3? 
29     i    3 

72    !?2| 

iji[:  7,7^ 

For  Purlins  not  exceeding   12"  Depth,  with  4l/< 
values  of  X  2  ,  X  3  ,  X4  ,  give  good  Results.       Y  5  to  Y  1  0' 

CLEARANCES  FOR 

"  Connection  Clearance,  the  following  assigned 
derived  therefrom. 

9  INCH  PURLIN 

A 

1/5  PITCH 

1/4 

PITCH 

3O°  PITCH 

1/3  PITCH 

65°  PITCH 

X 

X2= 

X3= 
6 

X4= 

X2= 

6% 

X3= 
6 

^ 

7 

5 

X4= 

X2= 

X3= 
5 

X4= 

X2= 

5 

4M 

Y    5 
Y    6 
Y    7 
Y    8 
Y    O 
Y1O 

3i7/82 

3'!''i,i 

3% 

8%2 

3% 

3%2 

725, 

1" 

2 

22^2 

2lB/g 

2% 

52%2 

52%2 

2^  ^^o 

2% 

»; 

2^4 

1% 

&7J.6 

CLEARANCES  FOR 

12   INCH  PURLIN 

A 

1/5  PITCH 

1/4 

PITCH 

30° 

PITCH 

1/3 

PITCH 

65°  PITCH 

X 

X2= 
7 

X3= 

X4= 

X2= 

w 

« 

X2= 
8 

X3= 

35 

X2= 

Xs  — 
7/2 

X4= 

1O 

Xs= 

^ 

Y    6 
Y    6 

Y    7 

Y  a 

Y    9 
Y1O 

8%2 

38y82 

9* 

9is/16 

36' 

82^2 

im. 

4%2 

3sy82 

3% 

82y82 

3% 

82y82 

?£ 

5%8 

82^2 

2% 

«: 

| 

14 

/ 

1% 

53/io 

n 

15 


SOLUTIONS,   FIVE   ORDINARY   ROOF    PITCHES 


A 

1/5  PITCH 

1/4  PITCH 

30°  PITCH 

1/3  PITCH 

55°  PITCH 

x 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log. 

Tan. 

"S" 

Log.  Tan. 

R 
L     1 

L    2 
L    3 
L    4 
L    5 
L     6 
L     7 
L    8 
L    9 
L  10 

21^6 

10% 

sy4 

3% 

9.45154 
9.56983 
9.96777 
9.38709 
9.95225 
9.93971 
9.25914 
9.29984 
9.96777 
9.43483 
9.43483 

5% 

108/4 

3% 

10^6 

10% 
4 
4 

9.54845 
9.65051 
9.95154 
9.45154 
9.92867 
9.91195 
9.33379 
9.39524 
9.95154 
9.52288 
9.52288 

4% 

928/82 

9.61092 
9.69897 
9.93753 
9.48599 
9.90853 
9.88908 
9.37641 
9.45593 
9.93753 
9.57745 
9.57745 

621,82 

10 

93/16 
823/32 

10 

5% 

9.67339 
9.74406 
9.92015 
9.51369 
9.88387 
9.86190 
9.41357 
9.51558 
9.92015 
9.62982 
9.62982 

6% 
4 

5% 

3% 
8i%2 

0.00426 
9.91336 
9.75859 
9.52144 
9.66984 
9.64710 
9.47851 
9.78984 
9.75859 
9.85160 
9.85160 

"  S"  =  Corresponding  Bevels  or  Slopes  to  Base  of  12  inches. 

A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

55°  PITCH 

X     1 

1  3 

A 

2^;  3    4K6K 

1K2K2   3    4M6M 

iH 

2H  3  WeM 

1^2^ 

1    9     -I  1  3 

8  *¥»  4r5 

-  H  H 
*af||8f| 

1  *M    6 

221 
3"  2" 

8ff 

iKafc 

J    3 

1  i 

3    4|f 

fl     o2  7 

Y    1 
Y    2 
Y    3 

Y    4 

H  H  iA.iH 

(L  ]    3   !    9  i   i  4 

A 

i-3 

21 
;( 

\     1 

[  i   -j9j- 

i  227 
|«X 

iH 

4 
6 

19 

2% 

ti 
»i 

3;', 

11       I/       23 
72      72      T2- 

A    il 

iHa> 

2^31 

11        19  ;     23 

For  Purlins  not  exceeding  12"   Depth,  with  4>£"  Connection  Clearance, 
values  of  X2,  X3,  X4  give  good  Results.     Y5  to  Yj  0  derived  therefrom. 

CLEARANCES  FOR  9  INCH  PURLIN 

the  following   assigned 

A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

55°  PITCH 

\7 

/\ 

X2= 
1 

X3= 

6% 

4y2 

™   fv, 

^ 

X2= 
8 

X3= 

6% 

X4= 

X2= 

X3= 

6V2 

44y; 

X2= 
10 

^ 

X4= 

4% 

Y    6 
Y    6 
Y    7 
Y    8 
Y    9 
Y1O 

6% 

7%, 

1019/aa 

m. 

71%o 

14%2 

12M.2 

5»^2 

58Ae 

,* 

S 

72%2 

72%S 

6% 

4y4 

6% 

4%2 

6% 

6% 

3%a 

6% 

I 

CLEARANCES  FOR  12  INCH  PURLIN 

A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

55°  PITCH 

8 

X3= 

X4= 

X2= 

X3= 

*; 

X2= 
9 

X3= 

^ 

9M 

X3= 

S 

X2= 

X3= 

X4= 

Y    5 
Y    6 
Y    7 
Y    8 
Y    9 
Y1O 

711/16 

8 

81y32 

13^6 

8%2 

ley* 

7%2 

101%2 

6i%6 

62^82 

59/82 
1019/82 

6% 

1019/82 

6% 

3%2 

6% 

411,62 

4y4 

6% 

6% 

16 


SOLUTIONS,  FIVE  ORDINARY  ROOF  PITCHES 

B=50° 


A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

55°  PITCH 

X 

"S" 

Log 

.  Tan. 

"S" 

Log 

.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

"S" 

Log.  Tan. 

R 

L     1 
L    2 
L    3 
L    4 
L    5 
L    6 
L     7 
L    8 
L    9 
L  10 

3%2 

13%3 

122%2 

lite 

21  »/32 

2% 

13%2 

3»/4e 
3»Ae 

9.41013 
9.64602 
0.04396 
9.49804 
0.02563 
0.00511 
9.33433 
9.29881 
0.04396 
9.47241 
9.47241 

6!%2 

4% 

121/32 

3%2 

3 

122%, 

4% 

9.50704 
9.72670 
0.02773 
9.56250 
0.00062 
9.97344 
9.41167 
9.39706 
0.02773 
9.56188 
9.56188 

7%2 

12% 
10% 

3^6 

12% 

9.56951 
9.77516 
0.01372 
9.59694 
9.97927 
9.94775 
9.45655 
9.46025 
0.01372 
9.61767 
9.61767 

•Mi 

112%2 
5^6 

3% 
4 

112%2 

5% 
6% 

9.63198 
9.82024 
9.99634 
9.62465 
9.95309 
9.91761 
9.49632 
9.52286 
9.99634 
9.67155 
9.67155 

11 

112%2 
5%2 

6% 
5% 

8 

9.96284 
9.98955 
9.83478 
9.63240 
9.72610 
9.68942 
9.57824 
9.82305 
9.83478 
9.90630 
9.90630 

"  S  "=Corresponding  Bevels  or  Slopes  to  Base  of  12  inches. 

A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

55°  PITCH 

X    1 

lM 

A 
A 

iff 

2M 

2} 

3/ 
fc 

3J 

4    3 

4 

M6M 

lM2M    3 

4^ 



ly9? 

A 

*H 

sM 

llA2l/2    3    4^6>i 

1^2^    3    4 

MQM 

2    2i^ 
H    IK 

ifcU 

4    3  J4J<6« 

Y    1 
Y    2 
Y    3 

Y    4 

i  i! 
i  A 

«3->S 

**H 

M  H 

^29  7  7 
*32!  '32" 

A  If  iA 
A  H  M 

H 

9f! 

I/              -iq               I/              11            -1 

1A  11  H  Is  is  5 

II  lAHiij 

A  ^|H; 

2.  A:  3>£  4f  2  6%  9  1| 

For  Purlins  not  exceeding  12"  Depth,  •with  4%"  Connection  Clearance,  the  following  assigned 
values  of  X2,  X3,  X4,  give  good  results.     Y5  to  Yi0  derived  therefrom. 

CLEARANCES   FOR  9  INCH  PURLIN 

A 

1/5  PITCH 

1/4  PITCH 

3O°  PITCH 

1/3  PITCH 

65°  PITCH 

\7 

X?= 

X5K 

X4= 

4/2 

X2= 
8 

«< 

X4= 

X2= 

X3= 

X4= 

X2= 
9 

X3= 

*£A 

J£ 

X3=      Xt= 

5M       4H 

Y    5 
Y    6 
Y    7 
Y    8 
Y    9 
Y10 

8% 

m. 

13%2 

14 

97<!2 

16% 

61*2 

6% 

6%2 

8^6 

61/82 

42T/00 

8%e     

Oyl.6 

7 

5%2 

7 

6%a 
7 

7 

31%6 

7 

CLEARANCES  FOR    12  INCH  PURLIN 

A 

1/5  PITCH 

1/4  PITCH 

30°  PITCH 

1/3  PITCH 

56°  PITCH 

X 

X:5= 

Xs= 

X 

X^ 
9 

Xs= 

X4= 

X2= 

X3= 

X4= 

X2= 
10 

X3= 

4p2 

X2= 

?M    ?H 

Y    5 
Y    6 
Y    7 
Y    8 
Y    9 
Y10 

13%2 

10%2 

14 

10% 

108V82 
15%6 

lOHso 

8% 

8%2 

11% 



19tte 

11% 

iSL 

^  

7 

6%2 

5V82 

7 

*157i0 

31%6 

7 

7 

HOPPERS,  BINS  AND  CHUTES 

FORMULAE  AND  GRAPHICS  FOR  SOLUTION  OF  ANGLES 


17 


On  developed  surface 


dice'' 


Plan 


EXPLANATION 

Inclined  surfaces  ai,bi,  ei,  andai,  bi,  di,  intersect  on  line  ai,  bi,  forming  dihedral  angle  measured 
by  angle  L11.     (See  Section  S-S.) 

Vertical  section  a,  b,  c,  (Section  D-D)  divides  the  dihedral  into  two  dihedrals,  of  which  L10  and 
L10'  are  respectively  the  complements. 

Angles  R,  C  and  C'  must  be  determined  from  design. 

Rectangular  bottom  with  irregular  top  will  produce  slightly  warped  side  surfaces,  see  dotted 
lines  for  this  condition. 


GRAPHICS 

Choose  any  point  f  in  line  a,  b 

Draw  f,  g,  i  a,  b 

Draw  g,  ni  l  a,  c 

Project  f  to  fi  in  plan 

Draw  fi,  mi  and  fi,  ni  in  plan 

Then  mi,  ni,  fi  is  plan  of  Section  S-S 

Revolve  f  to  h  about  g 

Project  h  to  hi  in  plan 

Draw  hi,  mi,  and  hi,  ni 

Then  hi,  mi,  ni  is  true  view  of  Section  S-S 


FORMULAE 
Tan  L10  —  Sin  R  Cot  C 
Tan  L10'  -  Sin  R  Cot  C' 
L11  =  180°  —  (L10  +  LW) 
Tan  L12  =  Sec  R  Tan  C  Sec  L10 
Tan  L12'  =-  Sec  R  Tan  C'  Sec  LW 
Cos  L12  —  Cos  R  Cos  C 
Cos  L12'  —  Cos  R  Cos  C' 


18 


PIPE  CONNECTION,  RESULTANT  OF  TWO  BENDS 


NOTES:— 

KNOWN  ANGLES  :— 

Angle  A  in  plane  of  profile 

Angle  B  in  horizontal  plane  or  plan 
THE  PROFILE  is  the  vertical   section   taken    thru 

the  center  of  the  pipe  line 
THE  LINE  P-P  is  perpendicular  to  center  line  of  pipe 

in  plane  of  profile 

ANGLES  TO  BE  SOLVED:— 

Resultant  Angle  X 

Detail  Angles  Y1  and  Y2 
FORMULAE  :— 

Cos  X  —  Cos  A  Cos  B 


Tan  Y1- 


Tan  Y2~ 


"Cot  A  Sin  B 
-  Tan  B  Cosec  A 


When  one  portion  of  pipe  is  horizontal. 


PIPE  CONNECTION,  RESULTANT  OF  TWO  BENDS 


19 


KNOWN  ANGLES:— 
Angle  A  i  in  plane  of  profile  (a; 
Angle  A2  in  plane  of  profile  (b) 
Angle  B  in  horizontal  plane  or  plan 

ANGLES  USED:— 


CosA2  SinB 

Cos  A2  Cos  B  Sec  CI  Sin  C2-J 


C2  —  C1—  Al 

Tan  Cs  -  „    Si.n  Al  —  rr-  —  Tan  Al  Sec  B 
Cos  A  I  Cos  B 

C4  —  A2—  C3 
ANGLES  TO  BE  SOLVED:— 

Resultant  Angle  X 

Detail  Angles  Y1  and  Y2 
FORMULAE  :- 

Cos  X=  Cos  Al  Cos  A2  Cos  B  +  Sin  Al  Sin  A2 


Tan  Y1  =  Tan  B  Cos  Ci  Cosec  C2 
Tan  Y2  -  Tan  B  Cos  Cs  Cosec  C4 


SECOND  CASE:- 

When  neither  portion  of  pipe  is  horizontal 


21 


Refer  to  Page  10;  a  c  =unity. 


Tangent  R 


Tan  ff=C2,_b2 

C2,    F2 

"=Tan  A 
02,  r2=cl,  rl 
=Sec  B 

•.TanR=TanA 


Sec   B 
=Taa  A  Cos  B 


Tangent  LI 


fl,Zl=dl,za 

=(d,  m)  Tan  B 
But  d,  m=Sin2A 

.-.fl,zl=Sin2  A  TanB 
f ,  c=d,  c 
=Sin  A 

lin2  A  Tan  B 


=Sin  A  Tan  B 


Tangent  L2 

Tan  L2=$f> 

gl,  za=d'l,  z2 

=(d,  m)  Tan  B 
But  d,  m=Sin2  A 

.'.  gl,  zs=Sin2  A  Tan  B 
b,  g=b,  d 

=(d,  c)  Tan  A 
=Sin  A  Tan  A 
,_Sin2  A  Tan  B 
•    Sin  A  Tan  A 
Sin  A  Tan  B 

Tan  A 
=Cos  A  Tan  B 


Tangent  L4 


Tangent  L7 

fl.kl 

fl,  kl=c,'f 
=d,  c 
=Sin  A 
a,  d=Cos  A 
r2,  d2=(a,  d)  Sec  L2 
=Cos  A  Sec  L2 
T2,  k=(r2,  d2)  Sec  R 

=Cos  A  Sec  L2  Sec  R 
el,  kl=(r2,  k)  Csc  B 

=Cos  A  Sec  L2  Sec  R  Csc  B 

Tan  ,  7 Sin  A 

L/    Cos  A  Sec  L2  Sec  R  Csc  B 
=Tan  A  Cos  L2  Cos  R  Sin  B 

1")  TanB  Cos  BCosZ.2 


=TanA(^r 
Tan  A  Sin  R  Tan  B  Cos  B  CosZ.2 

Tan  A  Cos  B 
=Sin  R  Tan  B  Cos  L2 

Tangent  L8 


nl,  kl=n,m 
=m,  w 

=(d,  m)  Cos  A 
=Sin2  A  Cos  A 
kl,  v=(dl,  kl)CotB 
=(d,  m)  Cot  B 
=Sin2  A  Cot  B 

A  Cos  A 


.Tan  L8= 


Sin2  A  Cot  B 
=Cos  A  Tan  B 

Tangent  L9 


dl,  Z4=d2,  j 

=d2,k 

jl,  zi=kl,  z- 

=(dl,  ZB)  Tan  B 
dl,  zs=(d2,  k)  Sin  R 
.'.Jlf  Z4=(d2,  k)  Sin  R  Tan  B 
2,  k)  Sin  R  Tan  B 


.Tan  ^ 


, 

=Sin  R  Tan  B 
Tangent  L10 


.Tan 


rl,  hl=T2,  h 

=T2,  d2 

=Cos'2  A  Sec  B  Sec  R 
rl,  el— Csc  B 

x_Cos2  A  Sec  B  Sec  R 

Csc  B 

_Cos2  A  Sin  B 
"Cos  B  Cos  R 
=Cos2  A  Tan  B  Sec  R 


8  is  any  point  on  TS,  02 
hoose  location  such  that  si  will  fall  at  a 
tl,  sl=SinB 
t,  t2=tl,  rl 

=(rl,  si)  Sin  B 
=Tan  B  Sin  B 
ul,  tl=u,  t 
=p,  t 

=(t,  72)  Sin  R 
=Tan  B  Sin  B  Sin  R 
•  Tan  /  ->n_Tan  B  Sin  B  Sin  R 
..T&nUO-  SinB 

=Tan  B  Sin  R 


22 


ANALYTIC     PROOFS 


Tangent  L3 


1. 

2. 

3. 

4. 

6. 
6. 

7. 

8. 
9. 


Tan  L3 


= 


J=Sln2  A 
C2=Tan  A 


81n2A 
— CosB 
.       z 
*-CosR 


Sin-A 


~CosB 
q=(b2,  02)  Sin  R 


10. 
11. 
12. 

13. 

14. 
16. 
16. 

17. 
18. 


=Tan  A  Sin  R 
Sin  A  Sin  R 


Cos  A 


r=l-d 


Sin2A 


Cos  B  Cos  R 


Sin  A  Sin  R 
Cos  A 


l=(e+z)SlnR 
Sin  R 


Sin  R 


Sln2A 


Sin  A  Sin  R 
Cos  A 


Sin  R 
CosB 


23 

ANALYTIC     PROOFS 


Sin2  A  Cos  A— Sin  A  Sin  R  Cos  B  Cos  R 

Cos  B~Cos  R~Cos  A 

19-  Sin  R 

CosB 

Sin2 A  Cos  A-Sln  A  Cos2R  Tan  R  Cos  B 

Cos  A  Cos2R  Tan  R 

Sin2A  Cos  A— Sin  A  Cos2R  Tan  A  Cos2B 

Cos  A  Cos2R  Tan  A  Cos  B 

_      A  Q4«2.        Sin2  A  Cos2R  Cos2B 

Cos  A  Sin  A ,-<-„  A 

22  - COS  A 

Cos  A  Cos2R  Sin  A  Cos  B 
Cos  A 


23. 


24. 


Cos2A  Sin2A-Sin2A  Cos2R  Cos2B 

Cos  A 

Cos  A  Cos^R  Sin  A  Cos  B 
Cos  A 

Cos2 A  Sln2A-Sln2A  Cos2R  Cos2B 
Cos  A  Cos2R  Sin  A  Cos  B 


25.       But,  re,  02  =  i/     1.    +  Tan2A 
V  Cos2B 


26. 


y  CosiiB+Cos2A 

_  ./Cos2 A + Sin2 A  Cos2B 
27>  - "         Cos2A  Cos2B 

no  V  Sin3A  Cos2B+Cos2A 

28-  =    — cosrsrcos^B — 


29.  Cos  R= 


30. 


'Sin2 A  Cos2B+Cos2A 

Cos  A  Cos  B 

Cos  A 


l/Sin2A  Cos2B+Cos2A 
Hence  by  substitution  in  No.  24. 


Sin2A  Cos2A-Sin2A      0<   ZA       08.  o.  ^  Cos2B 

31.  Tan  13=-  -y—      -  VSin2A  OoyB+Cos-A/  - 

008  A  Sin  A  COS  B 


Sin2  A  Cos2  A  (Sin2  A  Cos2B+Cos2A)-Sin2A  Cos2  A  Gos2B 

32.  =  _  Sin2A  Cos2B+Cos'2A  _ 

Cos3A  Sin  A  Cos  B 
Sin2A  Cos2B  +Cos2A 

oo  Sin  A(Sin2A  Cos2B+Cos2A)-Sin  A  Cos2B 

Cos  A  Cos  B 

Sin  A(Sln2A  Cos2B+Cos2A-Cos2B) 
~ 


ox 


Cos  A  dos~B 


o=  _  Sin  A  [Cos2B  (Sin2A-l)  +CosgA] 

Cos  A  Cos  B 

oft  _  Sin  A  [Cos2B(-Cos2A)  +Cos2A] 

Cos  A  Cos  B 

„«  Sin  A(Cos2A-CosaA  Cos2B) 

Cos  A  Cos  B 

Sin  A  Cos2A(l-Cos2B) 
a°-  Cos  A  C5s~lJ 

Sin  A  Cos2  A  Sin2B 
39-  =  —  Cos^rCos~B~~ 

4O.  =  Sin  A  Cos  A  Sin  B  Tan  B 


24 

ANALYTIC     PROOFS 


Tangent  L6—  Refer  to  Page  22. 

1. 

a=Cos  A 

2. 

b=Cos2A 

3. 

c=Cos  A  Sin  A 

4. 

.       1 
d=Sin  B 

Cos2A 

6. 

e=CosB 

6. 

f=V/d2+e2 

l/Si^B  Cos4A+Cos2B 

. 

Cos  B  Sin  B 

8. 

Let  M=\/Cos2B+Cos4A  Sin2B             (for  convenience) 

9. 
1O. 

Then  f-Cos  B  Sin  B 

11. 

,.  /r-~=2A  o<^2A  j.Sin2B  Cos4A+Cos2B 

—  *\  /OOS  A  oln  A  •  H  s  

\                                 Cos2B  Sin2B 

V  Cos2  A  Sin2A  Cos2B  Sin2B+Sin2B  Cos4A+Cos2B 

12. 

Cos  B  Sin  B 

13. 

Let  P=VCos?A  Sin2A  Cos2B  Sin2B+Sin2B  Cos4A+Cos2B 

14. 

Til  J-Ll-l        1-1                                                                                              1  IB 

Then  h-Cos  B  gin  B 

15. 

Sin  G4=£ 

Cos  A  Sin  A  Cos  B  Sin  B 

16. 

P 

17. 

Sin  G2=f 

Cos2A 

Cos  B 

M 

Cos  B  Sin  B 

19. 

Cos2A  Sin  B 

M! 

20. 

Cos  Ga=y- 

1 

O  1 

Sin  B 

21. 

M 

Cos  B  Sin  B 

Cos  B 

22. 

23. 

g=d  Cos  Qa 

25 

ANALYTIC     PROOFS 


24. 

25. 
26. 
27. 

28. 
29. 

30. 

31. 
32. 
33. 

34. 
35. 
36. 

37. 
38. 

39. 
40. 

41. 

42. 
43. 

44. 

45. 
46. 

_  (     1     ^  /Cos  B\ 

Cos  B 
=  ErSIn~E 
m  =  g  Sin  G* 
/    Cos  B  \  /Cos  A  Sin 

A  Cos  B  Sin  B>\ 

Cos  A  Sin  A  Cos2B 

P                       ) 

MP 
k  =  d  Sin  Ga 
(     1     \  /Cos2  A  Sin  B^ 

-^SinB/V          M          t 

Cos2A 

=     M 

s=V  k2+m2 

=  /  (^)%  (^^ 

Sin  A  Cos2B\2 

~Tvrp—    —) 

=  ^p-t/P2Cos2A+Cos4B  Sin2A 

Let  N=v/P2Cos2A+Cos4B  Sin2 

N  Cos  A 
Then  s=  —  5jp  — 

Sin  Gs  =  ^- 
Cos  A  Sin  A  Cos2B 

A             (for  convenience) 

M  P 

N  Cos  A 
Sin  A  Cos2B 

N 
Cos  Ga=g- 
CQS2A 

-     N  Cos  A 
P  Cos  A 
n  =  g  Tan(B-Ga) 

Cos  B  Sin  (B—  Ga) 

-  M  Sin  B  Cos  (B-Ga) 
Cos  B  (Sin  B  Cos  Ga-Cos  B  Sin  Ga) 

M  Sin  B  (Cos  B  Cos 

Ga+Sin  B  Sin  Ga) 

26 

ANALYTIC  PROOFS 


„       -,/SinjBCosB  _ 

47  \~  M  • 


63 
64 
65 


Cos  B  Cos2A  Sin  B 


Cos  B  (Sin  B  Cos  B—  Cos  B  Cos2A  Sin  B) 

M  Sin  B(Cos2B+Cos2A  Sin2B) 
Cos  B  (Cos  B—  Cos  B  Cos2  A) 
:    M(Cos2B  +  Cos2A 


Cos2B(l-Cos2A) 
60-  ~M(Cos2B+Cos2ASin2B) 

Cos2B  Sin2A 
~M(Cos2B  +  Cos2A  Sin2B) 

52.  p=m  Tan  Ga 

53.  t=n—  p 

54.  =n—  m  Tan  Gs 

55.  y=t  Cos  Ga 

56.  =(n—  m  Tan  Ga)  Cos  Gs 

m 

57.  v=Cos  Q8 

58.  w=t  Sin  Gs 

59.  =(n—  m  Tan  Gs)  Sin  Gs 

60.  x=v+w 

Al  ._-,  m^-  +(n-m  Tan  Gs)  Sin  Gs 

*•**•  OOS   Gs 

Statement  for  reduction 
62.  Tan  £0=JL 

(n-m  Tan  Gs)  (Cos  Gs) 


^  m — +(n-m  Tan  Gs)  Sin  Gs 
Cos  Gs 

(n-m  Tan  Gs)  Cos2Gs 

=m+(n-m  Tan  Gs)  Sin  Gs  Cos  Gs 


m+(n— m  Cos  Gs/  Sln  Gs  Cos  Gs 

(n  Cos  Gs-m  Sin  Gs)  Cos  Ga 
Je-  -m+(n  Cos  Gs-m  Sin  Ga)  Sin  Gs 

n  Cos2Gs— m  Sin  Gs  Cos  Ga 
~~m+n  Cos  Gs  Sin  Gs-m  Sin2Gs 

n  Cos2Gs-m  Sin  Gs  Cos  Q8 
~m  (1— Sin2Ga)+n  Cos  Gs  Sin  Gs 
_n  Cos2Gs-m  Sin  Gs  Cos  Gs 
~m  Cos2Gs+n  Cos  Gs  Sin  Gs 


27 

ANALYTIC     PROOFS 


__  n  Cos  Ga—  m  Sin  Ga 

~~m  Cos  Gs+n  Sin  Gs 
Hence  by  substitution 

/  Cos2B  Sin2A  \  /P  Cos  A\     /Cos  A  Sin  A  Cos2B\  /Sin  A  Cos2B\ 

71  Tan  /ff-\M(Cos2B+Cos2ASin2B)/\       N       /     \  _  MP  /  \          N          / 

/Cos  A  Sin  A  Cos2B\  (P  Cos  A\  .  /  Cos2B  Sin2  A  \/Sin  A  Cos2B\ 

\  MP  A       N       /     \M(Cos2B+Cos2A  SinaB)/\  N  j 

P  Cos2B  Sin2A  Cos  A  Cos  A  Sln2A 


_2  _  MN(Cos2B  +Cos2A  Sin2B)  _  MP  N 

P  Cos2A  Sin  A  Cos2B    +    _  Cos4B  Sin3A 

M  P  N  MN(Cos2B+Cos2A  Sin2B) 

=P2Cos  A  Cos2B  Sin2A-Cos4B  Sln2A  Cos  A  (Cos2B+Cos2A  Sin2B) 

P  Cos2A  Cos2B  Sin  A(Cos2B+Cos2A  Sin2B)  +P  Cos*B  Sin3A 
_          Sin2A  Cos2B  Cos  A  [P2-Cos2B(Cos2B+Cos2A  Sin2B 
P  Sin  A  Cos2B  [Cos2A  (Cos2B+Cos2A  Sin2B)  +  Cos  B  Si 
J3in  A  Cos  A  [P^Cos^  (Cos2B+Cos2A  Sin2B)] 
~  P  [Cos2A  Cos2B+Cos4A  Sin  B+Cos2B  Sin2A] 
J31n  A  Cos  A  [F*-CO8*B  (Cos2B+Cos2A  Sin2B)] 

P  [Cos2B  (Cos2A+Sin2A)+Cos4A  Sin2B] 

__  =Sin  A   Cos  A  [P2-Cos*B(Cos2B+CosiiA  Sin2B)] 

P  [Cos2B+Cos*A  Sin2B] 

78.  Cos  L4=% 


79. 

Sin  B^Cos  B 

80.  =    °p 

81.  .'.PCos /.4=CosB 

82.  r_  Cos  B 

83.  P2=Cos2A  Sin2A  Cos2B  Sin2B+Sln2B  Cos4A+Cos2B 

-SinACos  ACos/.4[Cos2ASin2ACos2B  Sin2B+Sin2BCos4A+Cos2B-Cos4B-Cos2BSin2B  Cos?A 

oft.  ian/.o — 5 T 5 

Cos  B  [Cos2B  +  Cos4A  Sin2B] 

_Sin  A  Cos  A  Cos  L4  [Sin2B  Cos2B  Cos2A  (Sin2A-l)+Cos2B  ( l-Cos2B)  +Sin2B  Cos4 A] 


85. 


Cos  B  [Cos2B+Cos4A  Sin2B] 
Sin  A  Cos  A  Cos  L4  [-Cos4 A  Sin2B  Cos2B+Cos2B  Sin2B+Sin2B  Cos4 A] 

T> 


Cos  B  [CosJB+Cos*A  Sin-fi] 
Sin  A  Cos  A  Cos  L4  [Cos4A  Sln2B  (l-Cos2B)+Cos2B  Sin2B] 
87 '  Cos  B  (Cos2B+Cos4A  Sin2B) 

Sin  A  Cos  A  Cos  Z.4(Cos4A  Sin4B+Cos2B  Sin2B) 
88.  = 


89. 


Cos  B  (Cos-B  +Cos*A  SinJB) 
Sin  A  Cos  A  Cos  L4  Sln2B  (Cos4A  Sln2B+Cos2B) 

Cos  B  (Cos2B+Cos4A  Sin2B) 
Sin  A  Cos  A  Cos  L4  Sin2B 


-  CosB 

91.   But,  Tan  L3=Sin  A  Cos  A  Sin  B  Tan  B 

Sin  A  Cos  A  Sln2B 
y^-  CosB 

93.        .'.  Tan  Z.ff=Cos  L4  Tan  L3 


28 


ANALYTIC     PROOFS 


Tangent  L.5 

Draw  d.2,  22 1  b2,  ca 

Pass  a  plane  thru  za  1  d2,  02 

This  plane  seen  in  plan  view  appears  as  surface  Z2,  ze,  ZT 

Revolve  this  plane  about  zs,  ZB  to  TA 

This  plane  then  seen  in  plan  view  appears  as  surface  Z2,  ze. 


2. 
3. 
4. 
6. 
6. 
7. 
8. 

9. 

10. 

11. 
12. 

13. 

14. 

15. 

16. 
17. 


Zl.   Z5=:Z2,  Z8 

=(z2,  d2>  Cos  Gl 
=(ml,  dl)  Cos  Gl 
=(d.  m)  Sec  B  Cos  Gl 
=Sin2A  Sec  B  Cos  Gl 
zl.  za=(d,  m)  Csc  B 
=Sin2A  Csc  B 

•  Tan  /  c—  Sln2A  Sec  B  Cos  Gl 
Sin2A  Csc  B 
Sln  B  Cos  Gl 


Cos  B 
=Tan  B  Cos  Gl 


But  Cos  GI= 


(Seepage  1O) 
Sln  A  Cos  A 
f,  c  Sec  L1 
ln  A  Cos  A 


~Sin  A  Sec  L1 
—Cos  A  Cos  L  1 
Hence  Tan  A5=Tan  B  Cos  A  Cos  LI 


18. 
19. 
2O. 
21. 

22. 

23. 
24. 
25. 

26. 
27. 

28. 
29. 
30. 


But  Cos   . 

C2> 

C2f  zs=C2.  T2  Sin  R 

_Sln  R 

Cos  B 

C2,  d2=Sin  A  Sec  L1 
Sin  R 

-  Cos~BI 
---- 


•   POR 
.  .  Cos 


Cos   Ll 

Sin  R  Cos  L1 


Sin  A  Cos  B 
And  Tan  £4=Cos2A  Tan  B  Sec  R 


COS  L3 


_Tan  R  Cos  L1  Cos  A  Tan  B 

Sin  A  CoifB 
_Tan  A  Cos  B  Cos  L1  Cos'2 A  Tan  B 

Sin  A  Cos  B 
_Sin  A  Cos  B  Cos  L1  Cos2A  Tan  B 

Cos  A  Sin  A  Cos  B 
=Cos  A  Tan  B  Cos  L1 
.'.Tan  L5=Cos  L3  Tan  L4 


PROOF  FOR  THE  90°  USED  JN  STY^ES^  AND  D 


29 


EXPLANATION    FOR  9O°   BEND  LINE   ON   STYLES   C  AND   D. 

Purlin  Web  Plane  seen  from  Elevation  of  Main  Roof  is  Line  cd. 

in  Plan  View  is  Inclined  Surface  dit  zi,  ci.  ki. 
Rafter  Lug  Plane  seen  in  Rafter  Elevation  is  Line  d2,  k. 

"   Plan  View  is  Inclined  Surface  di,   za,  ki. 
These  Two  Planes  produced  Intersect  on  Line  di,  ki. 
Hence  if  ki,  ci  equals  in  length  di,  zi,  then  will  angle  kt,  dii  zi  be  9O°  in  all  cases. 


STATEMENT  OF  VALUES. 

a  c=  Unity 

cd=Sin  A 

ad=CosA 

zl,  al=Cos2A 

zl,  ml=Sin'-A 

dl,  ml=Sin-'A  SecB 

zl.  dl=Sin-'A  TanB 

dl,  rl.rs,  Z4=Cos2A  SecB 

6.2,  r>=Cos2A  SecB  SecR 

da,  zt=d.  zs=CosA  Sin  A 

z*.  k=CosA  Sin  A  TanR 

da,  b2=Sin2A  SecB  SecR 

k,  c2=Sin2A  SecB— Cos  A  Sin  A  TanR 

kl,  cl=(k,  ca)  CscB 


=[Sin2ASecB-CosASinA  TanR]  Csc  B 


PROOF. 
1 .  (Sin*  A  Sec  B-Cos  A  Sin  A  Tan  R)  Csc  B=Sin2  A  Tan  B 


2.  CosB 


-Cos  A  Sin  A  Tan  R 


-=Sin2A  TanB 


SinB 
„    Sin2 A— CosASin  ACosBTan  R 

Cos  B"Bin  B  —  =Sin2A  Tan  B 

4.  Sin2A-Cos  A  Sin  A  CosB  Tan  R=Sin2  A  TanB  CosB  Sin  B 

5.  Sin2  A-Cos  A  Sin  A  CosB  Tan  R=Sin2   A  Sin2  B 

6.  Tan  R=Tan  A  Cos  B. 

7.  Sin*  A-Cos  A  Sin  A  Cos  B  Tan  A  Cos  B=Sin2A  Sin2B. 

8.  Sin  sA_CosASlnACos2B  SinA^.^  sln2B 

9.  Sln2A-Sin2A  Cos2B=Sin2A  Sin2B 

10.  I— Cos2B=Sin2B 

11.  l=Sin2B+Cos2B      or  1=1 

Hence  Eq.  1 1  being  true,  proves  Eq.  1  to  be  true. 


30 

ANALYTIC  .  PROOF    OF    ANGLE  X 


SECOND  CASE  OF  PIPE  LINE 


=C2  +  B2 
C    =  Sin  Al  +  Sin  Aa 
B    =^7(008  Al  +  Cos  Aa  Cos  B)  2  +  (Cos  An  Sin  B)2 


P    =v/(Sln  Al  +  Sin  Aa)2  +  (Cos  Al  +  Cos  A2  Cos  B)2+(Cds  Aa  Sin  B)2 

^(Sln^AI  +2  Sin  Al  Sin  A2  +Sln2Aa)  +(Cos2AI  +2  CosAI  CosAa  CosB  +Cos2A^  Cos2B)  +(Cos2A2  Sln^B) 
=1/Sln2AI+Cos2AI+Cos2A2(Sln2B+Cos2B)+Sln2A2+2  Cos  Al  CosAa  Cos  B+2  SlnAI  Sin  A2 


=^  I  +Cos2A2(l)+Sin2A2  +2  Cos  Al  Cos  A2  Cos  B+2  Sin  Al  Sin  Az 
=^/  2+2  Cos  Al  Cos  Az  Cos  B+2  Sin  Al  Sin  Aa 

Y  _  __  _ 

Cos—  =  \  V2  *  2  Cos  Al  Cos  Aa  Cos  B  +  2  Sin  Al  Sin  Aa 


=  2(2  *  2  Cos  Al  Cos  A2  Cos  B  +  2  Sln  Al  81n  Aa)     _  , 
4 

=  COS  Al  COS  Aa  Cos  B  +  Sin  A!  Sin  Aa 

When 

Al  or  Aa  =  O 
above  formula  becomes 
Cos  X  =  Cos  A  Cos  B 
which  Is  same  as  first  case 

The  following  formula  (proof  omitted)  was  developed  by  Mr.  C.  W.  L.  Filkins. 

_       y      SlnAjjCosCa 
Cos  *  —        SinCi 


Rl 


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